MultGate: A gate for multiplying parameters.

[ScottWe]

Summary

This pull request introduces a new gate which allows for parameters to be multiplied together. Due to limitations of Non-Linear Real Arithmetic, the python implementation requires that one of the parameters be an integer.

Details

In OpenQASM, it is possible to write parameterized gates whose arguments are written as linear sums of parameters. These are hard to express using only the AddGate and NegGate.

For example, consider the statement rz(5 * p[0] + p[1]) q[0];. To write this as a DAG in Quartz, the term 5 * p[0] must be decomposed into a sequence of AddGate node. For example, one could write:

[4, "", ["add", ["P1"], ["P0", "P0"]], ["add", ["P2"], ["P1", "P1]], ["add", ["P3"], ["P0", "P2"]]]

This pull request introduces a new gate which allows for parameters to be multiplied together.

Features

1. A new gate, MultGate, which takes two parameters wires and returns their product.
2. A python implementation for the gate, which handles the case where one of the wires is integral. The implementation achieves logarithmic runtime by applying the double-angle formula and triple-angle formula.

[xumingkuan]

Looks great! Thank you for your contribution.

[xumingkuan]

Sorry, what is NLRA?

[ScottWe]

Whoops: non-linear real arithmetic.

[xumingkuan]

I see. I prefer not using the acronym here :)

[xumingkuan]

Btw this seems not accurate: there is also half-angle formula:
sin(x/2) = ±sqrt((1-cos(x))/2)
cos(x/2) = ±sqrt((1+cos(x))/2)
The signs can be derived from the signs of sin(x) and cos(x).

[ScottWe]

Oh, cool. I didn't realize there was a z3.Sqrt function. Though I imagine the constraints would be really hard to solve when the parameters are symbolic.

In any case, I'll update the commit. Thanks!

[ScottWe]

@xumingkuan How were you thinking of determining sgn(cos(x/2)) from cos(x) and sin(x).

At first, I was thinking that sgn(sin(x/2)) = 1 and sgn(cos(x/2)) = z3.If(sin(x) >= 0, 1, -1). Then I realized that this is only true when x is in [0, 2pi]. For example, if x = pi/2 + 2pi, then x/2 = pi/4 + pi, and as a result sgn(sin(x)) = 1 =\= -1 = sgn(cos(x/2)).

Can we assume the verifier only works with angles from [0, 2pi]?

[xumingkuan]

Good question! I would also assume that the verifier only works with angles in [0, 2pi) but there are cases that the user needs to be aware of. For example, when computing 1.5 * x, (3 * x) / 2 and x + (x / 2) may have different results.

Unless you need non-integer multiples, I prefer only supporting multiplying integer values, with a comment that dividing by 2 is feasible but the sign needs to be determined. It might be better to have another HalfGate (with only one input x and one output x / 2) for the user to specify if they want (3 * x) / 2 or x + (x / 2) in the future.

[ScottWe]

Yeah, I think a HalfGate would make more sense.

Personally, I don't have a use-case for the HalfGate. I'm happy to implement it and open a new PR if you think it would be useful though.

[xumingkuan]

Actually, I don't think HalfGate will be useful -- it seems to me that it's just better to have the symbolic parameter y = x / 2 and use 2 * y whenever the user needs x.